Face of a convex set

Remarks

The empty set and $C$ are faces of $C$. A one-point set $\set{x}$ is a face exactly when $x$ is an extreme point of $C$. A face of a face of $C$ is a face of $C$.

Example: for the square $C = [0, 1]^2$, the edge $F = [0, 1] \times \set{1}$ is a face. It is convex. If a point of $F$ lies strictly between $x_1, x_2 \in C$, the second coordinates of $x_1, x_2$ are at most $1$ and average to $1$ with positive weights. So both equal $1$, and $x_1, x_2 \in F$.

Non-example: the lower half $F = [0, 1] \times [0, 1/2]$ of the same square is convex, but the point $(1/2, 1/2) \in F$ lies strictly between $(1/2, 0)$ and $(1/2, 1)$, and $(1/2, 1) \notin F$, so clause 2 fails.

If $H$ is a supporting hyperplane of $C$, then $C \cap H$ is a face, called an exposed face. Not every face is exposed: in the stadium, the convex hull of two unit discs centred at $(\pm 1, 0)$, the point $(1, 1)$ is a face but not an exposed face.